Logical Characterization of Weighted Pebble Walking Automata

International audienceWeighted automata are a conservative quantitative extension of finite automata that enjoys applications, e.g., in language processing and speech recognition. Their expressive power, however, appears to be limited, especially when they are applied to more general structures than words, such as graphs. To address this drawback, weighted automata have recently been generalized to weighted pebble walking automata, which proved useful as a tool for the specification and evaluation of quantitative properties over words and nested words. In this paper, we establish the expressive power of weighted pebble walking automata in terms of transitive closure logic, lifting a similar result by Engelfriet and Hoogeboom from the Boolean case to a quantitative setting. This result applies to a general class of graphs that subsumes all the aforementioned classes

Hybrid Branching-Time Logics

Hybrid branching-time logics are introduced as extensions of CTL-like logics with state variables and the downarrow-binder. Following recent work in the linear framework, only logics with a single variable are considered. The expressive power and the complexity of satisfiability of the resulting logics is investigated. As main result, the satisfiability problem for the hybrid versions of several branching-time logics is proved to be 2EXPTIME-complete. These branching-time logics range from strict fragments of CTL to extensions of CTL that can talk about the past and express fairness-properties. The complexity gap relative to CTL is explained by a corresponding succinctness result. To prove the upper bound, the automata-theoretic approach to branching-time logics is extended to hybrid logics, showing that non-emptiness of alternating one-pebble Buchi tree automata is 2EXPTIME-complete.Comment: An extended abstract of this paper was presented at the International Workshop on Hybrid Logics (HyLo 2007

On Pebble Automata for Data Languages with Decidable Emptiness Problem

In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is decidable. We also show that it is robust: alternating, nondeterministic and deterministic top view weak PA have the same recognition power. Moreover, this model is strong enough to accept all data languages expressible in Linear Temporal Logic with the future-time operators, augmented with one register freeze quantifier.Comment: An extended abstract of this work has been published in the proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science (MFCS) 2009}, Springer, Lecture Notes in Computer Science 5734, pages 712-72

Register Set Automata (Technical Report)

We present register set automata (RsAs), a register automaton model over data words where registers can contain sets of data values and the following operations are supported: adding values to registers, clearing registers, and testing (non-)membership. We show that the emptiness problem for RsAs is decidable and complete for the $F_\omega$ class. Moreover, we show that a large class of register automata can be transformed into deterministic RsAs, which can serve as a basis for (i) fast matching of a family of regular expressions with back-references and (ii) language inclusion algorithm for a sub-class of register automata. RsAs are incomparable in expressive power to other popular automata models over data words, such as alternating register automata and pebble automata

Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur

Pebble transducers with unary output

Boja\'nczyk recently initiated an intensive study of deterministic pebble transducers, which are two-way automata that can drop marks (named "pebbles") on their input word, and produce an output word. They describe functions from words to words. Two natural restrictions of this definition have been investigated: marble transducers by Dou\'eneau-Tabot et al., and comparison-free pebble transducers (that we rename here "blind transducers") by Nguy\^en et al. Here, we study the decidability of membership problems between the classes of functions computed by pebble, marble and blind transducers that produce a unary output. First, we show that pebble and marble transducers have the same expressive power when the outputs are unary (which is false over non-unary outputs). Then, we characterize 1-pebble transducers with unary output that describe a function computable by a blind transducer, and show that the membership problem is decidable. These results can be interpreted in terms of automated simplification of programs.Comment: 39 page

Translation from Classical Two-Way Automata to Pebble Two-Way Automata

We study the relation between the standard two-way automata and more powerful devices, namely, two-way finite automata with an additional "pebble" movable along the input tape. Similarly as in the case of the classical two-way machines, it is not known whether there exists a polynomial trade-off, in the number of states, between the nondeterministic and deterministic pebble two-way automata. However, we show that these two machine models are not independent: if there exists a polynomial trade-off for the classical two-way automata, then there must also exist a polynomial trade-off for the pebble two-way automata. Thus, we have an upward collapse (or a downward separation) from the classical two-way automata to more powerful pebble automata, still staying within the class of regular languages. The same upward collapse holds for complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machine can be, by using suitable inputs, simulated by a classical two-way automaton with a linear number of states (and vice versa), despite the existing exponential blow-up between the classical and pebble two-way machines

First-Order and Temporal Logics for Nested Words

Nested words are a structured model of execution paths in procedural programs, reflecting their call and return nesting structure. Finite nested words also capture the structure of parse trees and other tree-structured data, such as XML. We provide new temporal logics for finite and infinite nested words, which are natural extensions of LTL, and prove that these logics are first-order expressively-complete. One of them is based on adding a "within" modality, evaluating a formula on a subword, to a logic CaRet previously studied in the context of verifying properties of recursive state machines (RSMs). The other logic, NWTL, is based on the notion of a summary path that uses both the linear and nesting structures. For NWTL we show that satisfiability is EXPTIME-complete, and that model-checking can be done in time polynomial in the size of the RSM model and exponential in the size of the NWTL formula (and is also EXPTIME-complete). Finally, we prove that first-order logic over nested words has the three-variable property, and we present a temporal logic for nested words which is complete for the two-variable fragment of first-order.Comment: revised and corrected version of Mar 03, 201

The Arity Hierarchy in the Polyadic μ\mu-Calculus

The polyadic mu-calculus is a modal fixpoint logic whose formulas define relations of nodes rather than just sets in labelled transition systems. It can express exactly the polynomial-time computable and bisimulation-invariant queries on finite graphs. In this paper we show a hierarchy result with respect to expressive power inside the polyadic mu-calculus: for every level of fixpoint alternation, greater arity of relations gives rise to higher expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282